To complete the table of values for the function \( f(x) = \frac{3}{x - 2} \), we'll calculate \( f(x) \) for several chosen values of \( x \). Below is a set of \( x \) values and the corresponding \( f(x) \) values. We'll choose values around the vertical asymptote at \( x = 2 \) to observe the behavior of the function.
Let's choose the following \( x \) values: 0, 1, 2, 3, 4, 5.
Now we can calculate \( f(x) \):
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\( x = 0 \): \[ f(0) = \frac{3}{0 - 2} = \frac{3}{-2} = -1.5 \]
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\( x = 1 \): \[ f(1) = \frac{3}{1 - 2} = \frac{3}{-1} = -3 \]
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\( x = 2 \): \( f(2) \) is undefined (vertical asymptote).
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\( x = 3 \): \[ f(3) = \frac{3}{3 - 2} = \frac{3}{1} = 3 \]
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\( x = 4 \): \[ f(4) = \frac{3}{4 - 2} = \frac{3}{2} = 1.5 \]
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\( x = 5 \): \[ f(5) = \frac{3}{5 - 2} = \frac{3}{3} = 1 \]
Now, we can summarize the results in a table:
| \( x \) | \( f(x) \) | |---------|----------------| | 0 | -1.5 | | 1 | -3 | | 2 | Undefined | | 3 | 3 | | 4 | 1.5 | | 5 | 1 |
Now, let's put the points for \( x \neq 2 \) to graph the function:
- \( (0, -1.5) \)
- \( (1, -3) \)
- \( (3, 3) \)
- \( (4, 1.5) \)
- \( (5, 1) \)
Graphing Instructions:
- Plot the points in the Cartesian plane.
- Draw a vertical dashed line at \( x = 2 \) to indicate the asymptote.
- Connect the points smoothly, noting that as \( x \) approaches 2 from the left, \( f(x) \) goes to negative infinity, and as \( x \) approaches 2 from the right, \( f(x) \) goes to positive infinity.
The function will have a hyperbolic shape, located in the second and first quadrants away from the vertical asymptote.